Method for detecting displacements of the brain areas and the representation thereof on a display, said displacements caused by tumour formation

ABSTRACT

The invention relates to a method of detecting the displacement of individual brain areas and to the presentation thereof on a display for guaranteeing optimal planning of an operation, said displacements being caused by tumor formation. According to the invention a displacement vector is detected for each brain area which was situated at a defined location in the initial position where a tumor is located now. Said displacement vector defines the displacement of a brain area in the direction and the amount thereof, whereby said displacement is caused by tumor growth. Said vector is detected by means of two parameters (b and c) which can be derived from the size and the kind of the tumor. The displacement vectors are numerically calculated according to the formula u r =cr+br −2  after the two parameters have been detected. The image of the tumor as well as the detected displacements are subsequently inserted into a stereotactic atlas. The brain areas are displayed at the actual locations in the thus obtained image. Improved planning for carrying out operations can thus be guaranteed by means of said representation.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method of detecting brain area displacementsas a result of tumor formation and their presentation on an image screenfor purposes of ensuring optimum planning for an operation.

2. The Prior Art

In neurosurgery, the availability of efficient navigational devices bymedical technology that conventionally guided stereotactic operatingmethods on a broad clinical basis were abandoned in favor of navigatedinterventions guided by image presenting controls. Neurosurgeons have attheir disposal CD-versions of stereotactic atlases developed byTalairach and Schaltenbrand. Surgical access paths can now be monitoredby atlas series which can be correlated with image series. It is thuspossible to estimate risks even before an operation. The correlationalso makes possible improved planning and provides important decisioncriteria for the execution of an operation, in the search for importantanatomical land marks and physiological centers in the case of specialprocedures, e.g. Parkinson. The occurrence of brain tumors leads tospatial demands within the skull and, hence, to changes in pressure andlocal changes of brain areas.

The disadvantage of known atlases is that they do not take intoconsideration local changes, in particular local displacements of brainareas. Hence, they can be used on a limited scale only in surgeryseeking to remove tumors.

OBJECTS OF THE INVENTION

It is an object of the invention to provide a method by whichdisplacement of brain areas as a result of tumor formation may bedetected and presented on an image screen by correlation withstereotactic atlases.

Other objects will in part be obvious and will in part appearhereinafter.

SUMMARY OF THE INVENTION

In accordance with the invention the object is accomplished bydetermining a displacement vector for each brain area which initiallywas located at a defined position and is now occupied by a tumor. Thegrown tumor will have displaced the cell tissue surrounding it. At thesame time, this will have resulted in a deformation of the mass of thebrain in the environment of the tumor. A mathematical model of thisdeformation can be established by means of vector functions. Theposition of each point of mass of an area within the brain may bedescribed by a position vector in the predetermined coordinate system.As a result of the growth of the tumor the points of mass will changetheir position. This change in position may be described by a vectorfunction. Every displaced point of mass will then be defined by a newposition vector.

The displacement vector may be defined with sufficient accuracy by theequationu (r)= u (0)+ r φ(0)wherein φ represents the distortion tensor. The distortion tensor issymmetrical and is defined by the expression φ=grad u.

As a result of tumor growth the structure of the tissue within apredetermined area of the environment of the tumor will lose itsmechanical equilibrium. Within the cell tissue internal forces arisewhich are also defined as strains. The deformation occurs as a reactionto these strains. Thus, in addition to the distortion tensor φ thestrain tensor ψ is of decisive significance. In a state of equilibrium,outside of the previously mentioned area, the resultant of all internalstrains disappears, so that ψ=0. The correlation between the distortiontensor φ and the strain tensor ψ is given by Hooke's law.

Forces effective at the immediate surface, such as the displacement bythe tumor, are modeled by the following parameters. The displacementvector which in its direction and size defines the tumor induceddisplacement of a brain area is characterized by a first parameter c₁and a second parameter b₁ both of which influence the effect of thedisplacement of points of mass. The two parameters b₁ and c₁ are ameasure of the size of the tumor and of the effect of the tumor on itsenvironment and may be deduced from these two boundary conditions. Thesize of the tumor may be derived directly from an image of the tumorprovided by MRT or CT. The effect of a tumor on its environment which isdefined by type, size and degree of the tumor may be determined from animage of the environment of the tumor. The area in which the tumor hasan effect on its environment, i.e. where the tissue is not in a state ofequilibrium, is defined by the maximum effect of the tumor. At a certaindistance from the tumor corresponding to the maximum effect the positionvectors will not have been subject to a change in position on account ofthe tumor formation. The maximum effect of the tumor may also beselected from empirically derived comparative values which may have beendetected, for instance, on the basis of the type of tumor, the size ofthe tumor and other patient specific data. Subsequently, the individualdisplacement vectors for every effected point of mass will benumerically calculated on the basis of the equationu _(r) =cr+br ⁻².

Once the individual displacement vectors have been calculated, the imageof the tumor as well as the detected displacements which have occurredas a result of the tumor formation are inserted into a stereotacticatlas. In the presentation obtained in this manner, the brain areas willthen be depicted in their actual positions.

Improved planning for the execution of surgery may be ensured by thispresentation. It is also possible to use the presentation of the brainof a patient changed by the tumor formation for purposes of training forplanning surgery. Furthermore, the data calculated and correlated withthe stereotactic atlas may be used for controlling a surgical computer.

DESCRIPTION OF THE DRAWING

The novel features which are considered to be characteristic of theinvention are set forth with particularity in the appended claims. Theinvention itself, however, in respect of its structure, construction andlay-out as well as manufacturing techniques, together with other objectsand advantages thereof, will be best understood from the followingdescription of preferred embodiments when read in connection with theappended drawings, in which:

FIG. 1 is a schematic presentation of displacement vectors forindividual brain areas.

DESCRIPTION OF THE PREFERRED EMBODIMENT

The appended drawing is a schematic presentation of a skull 1 with atumor 3 developed in a brain 2. Every one of the brain areas representedby the points of mass P₁, P₂, P₃, P_(n) has been displaced from itsinitial location defined by the position vectors r ₁, r ₂, r ₃, r _(n),to a new position having the respective position vectors r ₁₁, r ₂₂, r₃₃, r _(nn). This change in position is covered by the vector function u(r). For a fixed point of mass P₁, P₂, P₃, P_(n), u (r) is also calleddisplacement vector. The displaced mass point P₁₁ is then defined by thenew position vector r ₁₁=r ₁+u(r ₁). The displacement vector for masspoint P1 may be stated with sufficient accuracy asu ( r ₁)= u (0)+ r ₁φ(0)where φ connotes the distortion tensor. The distortion tensor issymmetrical and is defined by the expression φ=grad u. As a result ofthe tumor formation the surrounding tissue structure has lost itsmechanical equilibrium. There are internal strains within the celltissue which react by bringing about a deformation. Therefore, inaddition to the distortion tensor φ, the strain tensor ψ is of decisiveimportance. In a state of equilibrium which exists at a finite distancefrom the center of the tumor 3 and which is defined by the maximumeffect of the tumor, the resultant of all internal strains disappears,hence, div ψ=0. The correlation between distortion tensor and straintensor is provided by Hooke's law. Accordingly,

$\Psi = {\frac{E}{1 + \sigma}\left( {\Phi + {\frac{3\;\sigma}{1 - {2\;\sigma}}\Phi\; I}} \right)}$where

-   E=elasticity module (Young's modulus);-   σ=lateral contraction coefficient (Poisson's ratio);-   I=the unity matrix; and

$\phi = {\frac{1}{3}{\sum\limits_{i}{\phi\;{ii}\mspace{14mu}{the}\mspace{14mu}{median}\mspace{14mu}{{distortion}.}}}}$

By inserting this expression into the equationdiv ψ=0the result will be2(1-σ)grad div u −(1−2σ)rot rot u =0.

Forces acting on the immediate surface, such as the displacement by thetumor 3, are modeled by the following parameters.

At the margin of the tumor 3 the displacement vector may be readdirectly; it corresponds to the radius R_(A) of the tumor 3. Proceedingfrom a single cell, i.e. the GO cell 5, the tumor 3 grew from size r_(o)to its currently observed size R_(A). At a sufficiently large distanceR_(M) from the tumor 3 the strain coincides with the brain pressure.This distance is defined as maximum effect 4. The phrase maximum effectis to be understood as a spatial range which is still affected by adisplacement brought about by the tumor 3. Outside of this range, thepresence of the tumor 3 causes no displacement of brain areas, and noshifting of points of mass occurs.

In the simplest case tissue displacement as a result of tumor growthtakes place in a radial direction only. Proceeding from coordinates of asphere, any shift in the φ and Θ direction disappears, and in the rdirection it is only dependent from the distance r of any given point ofmass P₁, P₂, P₃, P_(n) from the center of the tumor 3. Hence, rot u=0,and grad div u=0 also. Therefore, div u=const, the constant being about3 c. The definition of the divergence yields the equationu _(r) =cr+br ⁻²for a shift in the r direction.

On the basis of this, the distortion tensor as well as the strain tensormay be calculated. Thus, radial strain may be defined as

$\Psi_{\pi} = {{\frac{E}{1 - {2\sigma}}c} - {\frac{2E}{1 + \sigma}b\; r^{- 3}}}$

Taking into account the previously mentioned boundary conditions, theparameters b and c may be calculated on the basis of two equations.

The following values are assumed for the selected example:

-   -   R_(A)=20 mm size of tumor 3;    -   R_(M)=50 mm maximum effect 4;    -   r_(o)=30 □m size of starting cell 5;    -   E=7.16.10¹ Pa elasticity module;    -   σ=0.33 lateral contraction coefficient;    -   p_(Hirn)=1.172 kPa brain pressure

It is possible mathematically to calculate the parameters b and c on thebasis of the published value r_(o), a starter cell or GO cell 5, as wellas of the material constants E, σ, p_(Hirn) and of the determinedinstantaneous value R_(A) and the maximum effect R_(M) of the tumor 3.It is done by the following equations:

$\begin{matrix}{\Psi_{\pi} = {{{\frac{E}{1 - {2\sigma}}c} - {\frac{2E}{1 + \sigma}b\; R_{M}^{- 3}}} = P_{Hirn}}} & \lbrack 1\rbrack\end{matrix}$andu _(RA) =cr _(o) +br _(o) ⁻².  [2]

The constant b may be defined by means of the equation for the radialstrain ψ_(n). However, for the present purpose the radial strain must beset to equal the brain pressure, and the size R_(M) (radius of themaximum effect 4) has to be defined for this equilibrium. It is to beassumed that R_(M) depends upon the type of tumor since the effect ofthe tumor 3, i.e. the induced displacements depend significantly on thetype of tumor (e.g. its consistency). In other words, each type of tumorwill have to be described in terms of a characteristic maximum effect 4.The term maximum effect 4 is intended to mean a spatial area in whichthe tumor 3 still causes a displacement. The maximum effect 4 may alsobe determined by a visual evaluation of examinations (e.g. by MRT, CT).Outside of the boundaries of this areas the presence of the tumor doesnot lead to any displacement of brain areas. The maximum effect 4 mayalso be determined by automatic image recognition or image evaluationmethods. However, empirically determined values depending upon the typeof the tumor and the size of the tumor (degree) which may be used todefine parameter b, are also available. To this end recourse may be hadto a data base which takes into account the correlation between maximumeffect and size of tumor, degree of tumor, as well as the specificcondition of the patient such as, for instance, his sex, age, locationof the tumor, and a current health profile or treatment.

On the basis of equation [2] the value of parameter c is as follows:

$\begin{matrix}{c = {{\frac{2}{3}10^{3}} - {3.70410^{13}\mspace{14mu}{mm}^{3}{b.}}}} & \lbrack 3\rbrack\end{matrix}$

Proceeding from equation [1], parameter b may be determined by thefollowing steps, by first entering the value for parameter c fromequation [3]:

$p_{Hirn} = {{\frac{E}{1 - {2\sigma}}\left( {{\frac{2}{3}10^{3}} - {3.70410^{13}\mspace{20mu}{mm}^{3}b}} \right)} - {\frac{2E}{1 + \sigma}b\;{R_{M}^{- 3}.}}}$

A further transposition of this equation initially results in

$p_{Hirn} = {{\frac{E}{1 - {2\sigma}}\frac{2}{3}10^{3}} - {\frac{E}{1 - {2\sigma}}3.70410^{13\mspace{14mu}}{mm}^{3}b} - {\frac{2E}{1 + \sigma}b\; R_{M}^{- 3}}}$

followed by

$p_{Hirn} = {{\frac{E}{1 - {2\sigma}}\frac{2}{3}10^{3}} - {{b\left( {{\frac{E}{1 - {2\sigma}}3.70410^{13}\mspace{14mu}{mm}^{3}} + {\frac{2E}{1 + \sigma}b\; R_{M}^{- 3}}} \right)}.}}$

Therefore, the resulting equation for solving parameter b is:

$\begin{matrix}{b = {- \frac{p_{Hirn} - {\frac{E}{1 - {2\sigma}}\frac{2}{3}10^{3}}}{\left( {{\frac{E}{1 - {2\sigma}}3.70410^{13}\mspace{14mu}{mm}^{3}} + {\frac{2E}{1 + \sigma}\; R_{M}^{- 3}}} \right)}}} & \lbrack 4\rbrack\end{matrix}$

Parameter b having thus been calculated on the basis of the initialvalues, its value may be inserted into equation [3] to calculateparameter c. Once parameters c and b have been determined, eachdisplacement u_(r) for each individual point of mass P₁, P₂, P₃, P_(n)which in its initial state, i.e. before tumor 3 developed, was locatedat a position within R_(M), may be unambiguously calculated.

The displacement vectors u(r) for each point of mass P₁, P₂, P₃, P_(n)thus calculated are determined by suitable software in order to obtainthe new position vectors r₁₁, r₂₂, r₃₃, R_(nn). The new coordinates forthe displaced points of mass P₁₁, P₂₂, P₃₃, P_(nn) are inserted intoexisting stereotactic or digitized anatomic atlases to makevisualization possible of the changes of brain areas resulting from thetumor formation.

It is thus possible during the planning stages of surgery to takedisplacements of areas into account. It is then that meaningful use maybe made of the correlation of anatomic atlases and image series (CT/MRT)of an infected patient.

1. A method of determining displacements of brain areas as a result of tumor formation and of their presentation on an image screen utilizing stereotactic atlases characterized by determining by means of known physical correlations, particularly Hooke's law, a displacement vector u(r) for each point of mass of a brain area (P₁, P₂, P₃, P_(n)) which in an initial state was located at a position described by a position vector (r₁, r₂, r₃, r_(n)) where the tumor (3) is now located, which displacement vector may be unambiguously defined by two parameters b, c, whereby the parameters b, c may be derived from the boundary conditions tumor size and tumor type and the displacement vectors u(r) are subsequently mathematically calculated and a new position vector (r ₁₁, r ₂₂, r ₃₃, r _(nn)) is calculated for each point of mass of a brain area (P₁, P₂, P₃, P_(n)) as a result of the displacement, thereafter inserting the image of the tumor (3) as well as the determined displacements with the new position vectors (r ₁₁, r ₂₂, r ₃₃, r _(nn)) which took place as a result of the tumor formation, into an anatomic atlas so that the brain areas characterized by the displaced points of mass (P₁₁, P₂₂, P₃₃, P_(nn)) are now displayed in the actual positions.
 2. The method of claim 1, characterized by the fact that the parameters b, c are determined from an image of the tumor (3), in particular the size of the tumor and from an additional evaluation of the effects of the tumor formation by determining the maximum effect on the environment of the tumor (3).
 3. The method of claim 1, characterized by the fact that the effect of the tumor formation on the environment of the tumor (3) which is characterized by the maximum effect, is determined from empirically determined correlations depending upon the type of tumor, the size of the tumor and further patient specific data.
 4. The method of claim 3, characterized by the fact that the parameters b, c are determined by image recognition methods.
 5. The method of claim 1, characterized by the fact that the displacement vectors u(r) for each point of mass of a brain area (P₁, P₂, P₃, P_(n)) are determined by the equation U_(r)=cr+br⁻². 